Question

In the illustration below, the three cube-shaped tanks are identical. The spheres in any given tank

are the same size and packed wall-to-wall. If each of the tanks are filled to the top with water, which
tank would contain the most water. Prove your answer algebraically using x to represent the edge
length of the tanks.

Answer 1

Explanation:

Let represent the edge of the tank with x and the radius of the first sphere with x/2;

The amount of the water = Volume of the tank - Volume of the sphere

=
`x^3 - (4)/(3) \pi ((x)/(2))^3`

on the second cube ; the radius of the sphere =
`(x)/(4) \ units` ;

Also the number of sphere here is = 8

The amount of water =
`x^3 -8*(4)/(3) \pi ((x)/(4))^3`

For the third figure ; the radius of the sphere is =
`(x)/(8) \ units`

Also the number of sphere here is = 64

The amount of water =
`x^6 -64*(4)/(3) \pi ((x)/(8))^3`

=
`x^3 - (4)/(3) \pi ((x)/(2))^3`

In the fourth tank ; 512 sphere illustrates that in a single row; that more than one 8 sphere is present i.e 8³ = 512

then the radius will be =
`(x)/(16)`

The amount of water =
`x^3 -512*(4)/(3) \pi ((x)/(16))^3`

=
`x^3 -(4)/(3) \pi ((x)/(2))^3`

This implies that alll the three cube shaped tanks are identical and hold equal amount of water.